Zeno's pet rabbit always jumps halfway to a carrot no matter how far away he is from the carrot. He will eat the carrot if he lands within 6 inches of it. How many times must Zeno's rabbit jump to eat a carrot that is initially 12 feet away?
We can start by simply charting out how far away Zeno's rabbit is from the carrot after each jump. Once the rabbit is less than 6 inches away, he'll start chomping away at his treat. \begin{tabular}{c|clll} {Jumps} & {Distance} & & & \\ \cline{1-2} 0 & 12 feet & & & \\ 1 & 6 feet & & & \\ 2 & 3 feet & & & \\ 3 & 18 inches & & & \\ 4 & 9 inches & & & \\ 5 & 4.5 inches & & & \end{tabular} Thus, it takes Zeno's rabbit $\boxed{5}$ jumps to get to the carrot. Alternatively, note that the rabbit must reduce the original distance by a factor of $\frac{12\text{ ft}}{0.5\text{ ft}}=24$. He can cut his distance in $\frac{1}{2}$ after each jump, so after each $x$ hops, he is $\left(\frac{1}{2}\right)^x$ of the original distance away. The goal is to reduce the distance to at least $\frac{1}{24}$ of the original, so we seek the smallest $x$ such that $\left(\frac{1}{2}\right)^x>\frac{1}{24}$. Rearranging, we see that $2^x>24$, so the smallest number of hops $x$ must be $\boxed{5}$.